Fortunately, this backreaction problem has already been addressed in 2000 by Anamaria Font and one of us [1]}.
Similar to the solution for the backreacted original Sugimoto model [2]}, it was shown in [1]} that the gravity and dilaton equations of motion of the T-dual setup do not admit a solution with nine-dimensional Poincaré symmetry (along the longitudinal directions of the brane), but induce a spontaneous compactification on a transversal circle times a non-trivial longitudinal direction. In [1]}, two explicit solutions were presented, where for the first the longitudinal direction was non-compact, whereas for the second it was a finite size interval.
